Step 2:
In this step, a Delaunay triangulation is performed, using the points resulting from step 1. The triangulation process
becomes much faster due to the discretization of the cloud points made in step 1. In addition, those points ensure the creation of a
surface much closer to the reality of the terrain, eliminating incongruent situations, namely when two or more points have the
same planimetric X, Y coordinates, but different values of Z. The Laplacian operator is perhaps the most known and used
smoothing operator Belkin et al., 2008. This operator is based in the establishment of a neighborhood relation through the
triangles with shared edges. Figure 5 presents an example of the establishment of a neighborhood, where the red points represent
the neighborhood of the blue point.
Figure 5. Neighborhood relation through the shared edges. The original version of the Laplacian algorithm is quite simple:
the position of the blue point and the corresponding triangle vertex is replaced by the average point of the
neighbor’s positions red points Vollmer et al., 1999.
In this work, instead of changing the point position, the point is removed from the final DTM. Different operators are suggested
based on the defined neighborhood. For example, the TriMin and TriMax values Figure 5 represent the corresponding minimum
and maximum lengths of the edges established for the neighborhood.
Another operator to be used in this work is based on the plane or planes generated by the neighboring points Figure 6.
Figure 6. Example of the minimum distance D from a point to the plane defined by three neighboring points.
In Figure 6, the distance D represents the minimum distance between the blue point and a plane generated by the tree
neighboring points red points. In order to calculate D it is necessary to determine the plane equation from the three-
dimensional coordinates of the points. Considering the general equation of a plane as ax + by + cz + d =
0, the a, b, c and d coefficients can be obtained from a vector normal to the plane. The normal vector is computed from the
cross-product of two vectors connecting the points. Then, the distance D is calculated through Equation 4, where X
, Y , Z
are the blue point coordinates. � =
+ +
+ √ +
+ 4
In the case where a point has more than three neighbors, it is likely that not all stand on the same plane. In this case, statistical
methods can be applied to obtain the mean plane, namely by least squares or by defining all the planes that are generated by
grouping the neighbors three by three and calculating all the minimum distances between the generated planes and the point.
In Figure 7, the values PMax and PMin represent the corresponding minimum distances to the furthest and the closest
planes generated by the three grouped neighbors.
Figure 7. Minimum and maximum plan distance. The goal of using these operators is to eliminate all conspicuous
points by smoothing the surface generated by the points resulting from step 1, and to eliminate the points in quasi-plane areas,
which are not relevant for the representation of the terrain. The step 2 is applied through an iterative process. Two values of
distance, D
1
and D
2
are established. In each iteration a new Delaunay triangulation is created, and the points where D
1
PMax or D
2
PMin are eliminated. D
1
and D
2
may be adjusted according to the desired result. Taking into account that
neighborhood is reciprocal i.e., a certain point belongs to the neighborhood of each of the neighboring points, in order to avoid
an erosion of the model, a constraint is applied in each iteration. Based on that constraint, if any neighbor of a point is identified
to be eliminated from the model, that point cannot be eliminated in that iteration
, regardless of satisfying the condition ‘D
1
PMax or D
2
PMin’. The stopping criteria for the iterations can be established by an absolute minimum value for the number of
eliminated points or by setting a minimum value for the eliminated points between consecutive iterations.
It should be noted that the resulting points from Step 1 and Step 2 are points directly measured from the clouds, with no
interpolation process.
3. RESULTS
In order to test the different steps of the proposed method, C algorithms were implemented. This section depicts the obtained
results by the application of the two steps in point clouds collected by MLS. A sensitivity analysis of the proposed method
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292
parameters is presented. The following examples data were also chosen to demonstrate the application of the method in rural and
urban environments. 3.1
Sensitivity analysis
Figure 8a shows a point cloud sample in a rural area with trees. Figure 8b presents the result obtained by applying Step 1 to the
sample cloud of Figure 8a. It can be verified that the cloud points representing the vegetation, the house wall, electric power lines
and poles were globally eliminated, keeping only points at the terrain level. It is also observed that the resulting points do not
have a regular density, being the density smaller in plane areas, for example along the road, and higher in areas with greater
terrain variation.
Figure 8. a Point cloud example colored by height b Resulting points obtained from Step 1 application after four iterations,
where D
Ini
= 1 m, LMin = 0.04 m, LMax = 0.08 m. The results obtained by the variation of the initial cell dimension
D
Ini
are presented Figure 9. The remaining parameters values are the same as used in Figure 8b. By decreasing the D
Ini
value, the resulting points
’ density increases, allowing a better representation of the terrain details.
Figure 9. a Point cloud in urban area colored by height b Step 1 result with D
Ini
= 4 m c Step 1 result with D
Ini
= 2 m d Step 1 result with D
Ini
= 1 m. One of the great challenges in DTM creation and representation
is the modelling of the terrain break lines, being the line along the curb in urban areas a good example of this. In Figure 10, a
cross section of a point cloud crossing a curb is presented. The cross section has a width of 0.25 m and a length of 2 m. It is also
presented with red points the obtained result as the number of iterations of Step 1 increases. In the first iteration Figure 10a it
is only observed the points with the lowest Z value PZMin inside the initial cells. As the number of iterations increases
Figures 10b and 10c, the division of each of the initial cells allows a greater discretization of the terrain and consequently
increases the number of points.
Figure 10. a Step 1 result with one iteration b Step 1 result with two iterations c Step 1 result with three iterations.
By increasing the number of iterations, a greater concentration of points takes place in the break line area of the curb, while in the
horizontal areas the density does not increase. This effect results from the fact that in the following iterations, in the quasi
horizontal areas, none of the points inside the cells satisfy the constraint of Equation 2, and the process is stopped inside these
cells. In the areas of greater terrain variation the cells continue to divide until there are no more points that satisfy the condition or
until the stop criteria is reached. This explains the fact that the distance between the resulting points is inversely proportional to
the variation of the terrain. Parked cars along roads are another typical issue in DTM creation
from point clouds collected in urban environments. Figure 11a shows a point cloud where two parked cars were captured by the
MLS data collection. Figures 11b and 11c compare the obtained results of step 1 by variation of LMax value. By decreasing the
LMax value Figure 11c, the number of points representing the vehicles in the final result also decrease. That happens because
the LMax value used in the example shown in Figure 11c is less
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293
than the distance between the car floor and the ground. Thus, the step1 applied to the cells containing the points from the car stops
before reaching those points. The only points representing the car that remain in the Figure 11c are the points from the car wheels
that are in contact with the ground. In Figure 11b, the points representing the car floor that are collected directly or through
the car window remain in the result. The result presented in Figure 11c turns much easier the elimination of the car points in
step 2. In the Figure 11b result, it will be more difficult to eliminate those car points and at the same time keeping, for
example, the points representing the curb.
Figure 11. a Point cloud sample, colored by elevation b Step 1 result with LMax = 0.40 m c Step 1 result with LMax = 0.15 m.
In the application of Step 2, the S-hull algorithm Sinclair, 2010 was selected to implement the Delaunay triangulation, among the
various algorithms described in the literature. Figure 12a shows a hill shade representation of the Delaunay
triangulation based in the points of Figure 8b. Although the surface appears to make a correct representation of
the terrain, if an amplified zone is observed Figure 12b, it can be verified that there are many peaks particularly in the
undergrowth areas. This effect will be decreased by the application of step 2.
Threshold values of PMax and PMin are defined. The goal is, on one hand, the elimination of conspicuous points and smoothing
the surface and, on the other hand, the decrease of the number of the DTM points, keeping the terrain details, namely, the break
lines along the terrain. Figure 12. a Hill shade representation of the Delaunay
triangulation of Figure 8a result b Amplified red area of Figure 12a.
Figure 13 represents the same amplified area of Figure 12b, after the application of step 2. The result was obtained with the
restrictions PMax 0.05 m and Trimax 0.75 m. That means that in each iteration all points that satisfy both conditions are
eliminated. Besides that, in each iteration, no point can be eliminated if any of its neighbors are identified to be eliminated.
The number of points have been reduced from approximately 122000 in Step 1 result Figure 8b to 53000 points a reduction
of 57. The presented result was obtained after 37 iterations and took 42 seconds.
Figure 13. Result obtained after Step 2 application with PMax 0.05 m and Trimax 0.75 m.
Together with the smoothing process, the PMin value was used to decrease the number of points in flat areas. Figure 14 shows
the difference between the Delaunay triangulation of the Step 1 result Figure 14 a and the Step 2 result using PMin 0.005 m
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294
and TriMin 0.5 m Figure 14 b. Despite the fact that a lower point density can be observed in flat areas, the curb line is still
well defined.
Figure 14. a Delaunay triangulation of Step1 result b Delaunay triangulation of Step1 result and PMin 0.005 m and TriMin
0.5 m. Using the parameters of Figure 13 and Figure 14b on step 2, it
was possible to decrease the number of points to approximately 8000. Starting from the point cloud shown in Figure 8a, with
1238000 points, it took around 70 seconds in total to run both steps 1 and 2. The resulting data set has a decrease of 99 of the
initial points that are comprised of non-ground and redundant points. The remaining data set includes points that are part of the
terrain and are enough to create a realistic DTM. 3.2
Case studies of Angola and Brazil
The proposed method was applied in two MLS data collections along roads in different areas of the globe, Brazil and Angola.
Both are developing countries with huge infrastructures needs and the corresponding need of basic geographic information to
support their project design, namely DTM. In the Angola study case, it is intended to obtain a DTM from a rural road in Luanda
surroundings. The purpose of the DTM is to be used as a basis to the project design of a road asphalting. The data from Brazil was
collected in an urban neighborhood of Rio de Janeiro, and its purpose is the project design of a sanitation network.
The collected point clouds were obtained by different systems and processed by different software applications. In both cases
the clouds were exported to LAS 1.2 file format Specifications in Table 1.
Country MLS LAS number Lenght
Angola Trimble MX8
8 12 Km
Brazil Optech Lynx
19 23 Km
Table 1. Main data collection specifications The LAS number in Table 1 corresponds to the number of the
standard LAS files obtained in each data collection. The files can be split by the operator in the vehicle at the collection moment or
in the cloud processing software. First, the step 1 and step 2 were applied in sequence to each LAS
file independently. For each LAS file an ASCII file was created with the resulting points. That allows to reduce the number of
points in a way to subsequently merge the ASCII files. Step 2 is again applied to the merged ASCII file to get the DTM points for
the entire collected area.
Figure 15. Overall view of the Brazil DTM final result. The final DTM comprise approximately 248000 points in Brazil
Figure 15 and 65000 in Angola. Figure 16 shows a sample area of the Angola final result. The
DTM is represented by contours with 0.5 m distance and elevation points. The presented elevation points were chosen
directly from the resulting DTM points with no interpolation.
Figure 16. Angola data set area. Table 2 describes the processing time for each task of the process.
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295
Country Data
collection Cloud
Processing DTM
Processing
Angola 2 hours
6 hours 5 hours
Brazil 6 hours
13 hours 11 hours
Table 2. Tasks processing time. The planning time, travel, GPS reference establishment, control
points measure and data transfer are not included in Table 2, but even so it can be used to compare the different tasks processing
time and to show that the proposed method can be applied without having the processing time as a bottleneck of the entire
process.
4. CONCLUSIONS AND FUTURE WORK
